An extension of a theorem of Wu-Yau

By Valentino Tosatti, Xiaokui Yang. 2015

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We show that a compact Kahler manifold with nonpositive holomorphic sectional curvature has nef canonical bundle. If the holomorphic sectional curvature is negative then it follows that the canonical bundle is ample, confirming a conjecture of Yau. The key ingredient is the recent solution of this conjecture in the projective case by Wu-Yau.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/fe61319a1db41d2190d8ba22d836193193048945

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

RC-positivity, rational connectedness and Yau's conjecture (2017, background): “In [41], Tosatti and the author proved that any compact Kähler manifold with nonpositive holomorphic sectional curvature must have nef canonical line bundle, and with that in hand, we were able to drop the projectivity assumption in the aforementioned Wu-Yau Theorem.”
K\"ahler manifolds of semi-negative holomorphic sectional curvature (2014, background): “ifolds. Remark7.2. After this paper had been completed and gone to press at the Journal of Diﬀerential Geometry, Wu and Yau gave a proof of Conjecture 1.2 in [WY15]. Subsequently, Tosatti and Yang in [TY15] extended that proof to the K¨ahler case. However, since these new proofs, which are based on ”
On real bisectional curvature for Hermitian manifolds (2016, background): “o one has Ric(ωε) ≥ −ωε + ε0 +ε D ωh. Now if we apply the Schwarz Lemma Corollary 4.4 to the identity map from(Mn,ωε) onto (Mn,h), with λ= 1, µ= ε0+ε D , and κ= 0, then we get trωεωh ≤ nD ε0 +ε as in [16]. So trωεω≤ nD 2 ε0+ . The rest of the argument is identical to that of [16], so Theorem 1.7 hol”